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Lesson Worksheet: Linear Transformation Composition Mathematics

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In this worksheet, we will practice finding the matrix of two or more consecutive linear transformations.

Q1:

Let the matrix 𝐴 represent rotation in the plane through an angle of 𝜃 and let the matrix 𝐵 represent reflection in the 𝑥-axis.

What is the matrix 𝐴?

  • A𝜃𝜃−𝜃𝜃cossinsincos
  • B𝜃𝜃𝜃𝜃cossinsincos
  • C𝜃−𝜃𝜃𝜃sincoscossin
  • D𝜃−𝜃𝜃𝜃cossinsincos
  • E𝜃𝜃𝜃𝜃sincoscossin

What is the matrix 𝐵?

  • A−100−1
  • B−1001
  • C0110
  • D0−1−10
  • E100−1

What is the matrix 𝐴𝐵?

  • A𝜃𝜃𝜃−𝜃cossinsincos
  • B𝜃−𝜃𝜃𝜃cossinsincos
  • C𝜃−𝜃𝜃𝜃sincoscossin
  • D𝜃𝜃−𝜃𝜃cossinsincos
  • E𝜃𝜃𝜃−𝜃sincoscossin

Q2:

Suppose that the matrix 𝐴 represents rotation about the origin through an angle of 𝜃 (measuring between 0∘ and 90∘) and the matrix 𝐵 represents reflection in the 𝑥-axis.

Find 𝑀=𝐴𝐵.

  • A𝑀=𝐴𝐵=−𝜃𝜃−𝜃𝜃cossinsincos
  • B𝑀=𝐴𝐵=−𝜃−𝜃𝜃−𝜃cossinsincos
  • C𝑀=𝐴𝐵=𝜃𝜃𝜃−𝜃cossinsincos
  • D𝑀=𝐴𝐵=𝜃−𝜃𝜃−𝜃cossinsincos
  • E𝑀=𝐴𝐵=𝜃−𝜃𝜃𝜃cossinsincos

Note that 𝑀 is a reflection in a line through the origin. Let this line of reflection have equation 𝑦=𝑘𝑥. By considering the image of the vector ⟨1,0⟩, determine the measure of the angle between the 𝑥-axis and the line of reflection.

  • A𝜋2−𝜃
  • B𝜃2
  • C−𝜃2
  • D𝜃
  • E−𝜃

What, therefore, is the slope 𝑘 of the line of reflection?

  • Atan𝜃
  • Btan𝜃2
  • C−𝜃tan
  • D−𝜃2tan
  • Ecot𝜃

If v lies in the direction of the line of reflection, what is 𝑀v?

  • A⟨1,0⟩
  • B−v
  • C⟨0,1⟩
  • Dv
  • E⟨0,0⟩

By solving the equation obtained in the previous part, find another expression for the slope 𝑘 of the line of reflection.

  • A𝑘=𝜃1+𝜃=1−𝜃𝜃sincoscossin
  • B𝑘=𝜃1−𝜃=1+𝜃𝜃cossinsincos
  • C𝑘=𝜃1−𝜃=1+𝜃𝜃sincoscossin
  • D𝑘=𝜃1−𝜃=1−𝜃𝜃sincoscossin
  • E𝑘=𝜃1+𝜃=1−𝜃𝜃cossincossin

Q3:

Suppose 𝐴 and 𝐵 are 2×2 matrices, with 𝐴 representing a counterclockwise rotation of 30∘ about the origin and 𝐵 representing a reflection in the 𝑥-axis. What does the matrix 𝐵𝐴 represent?

  • AA reflection in the line through the origin at a 75∘ inclination
  • BA reflection in the line through the origin at a −15∘ inclination
  • CA reflection in the line through the origin at a −75∘ inclination
  • DA reflection in the line through the origin at a −45∘ inclination
  • EA reflection in the line through the origin at a 15∘ inclination

Q4:

Suppose 𝐴 and 𝐵 are 2×2 matrices, with 𝐴 representing a counterclockwise rotation of 30∘ about the origin and 𝐵 representing a reflection in the 𝑥-axis. What does the matrix 𝐴𝐵 represent?

  • AA reflection in the line through the origin at a −15∘ inclination
  • BA reflection in the line through the origin at a −75∘ inclination
  • CA reflection in the line through the origin at a 15∘ inclination
  • DA reflection in the line through the origin at a 75∘ inclination
  • EA reflection in the line through the origin at a −45∘ inclination

Q5:

Describe the geometric effect of the transformation produced by the matrix 0−330.

  • AA dilation with center at the origin and scale factor 3 followed by a reflection in the line 𝑦=−𝑥
  • BA dilation with center at the origin and scale factor 3 followed by a 180∘ rotation about the origin
  • CA dilation with center at the origin and scale factor 3 followed by a reflection in the line 𝑦=𝑥
  • DA dilation with center at the origin and scale factor 3 followed by a 90∘ rotation about the origin
  • EA dilation with center at the origin and scale factor 3 followed by a −90∘ rotation about the origin

Q6:

Which of the following compositions of transformations is represented by the matrix 0−2−20?

  • AA rotation of 180∘ about the origin followed by a reflection in the line 𝑦=−𝑥
  • BA rotation of 180∘ about the origin followed by a reflection in the line 𝑦=𝑥
  • CA dilation with a center at the origin and scale factor 2 followed by a reflection in the line 𝑦=𝑥
  • DA dilation with a center at the origin and scale factor −2 followed by a reflection in the line 𝑦=−𝑥
  • EA dilation with a center at the origin and scale factor −2 followed by a reflection in the line 𝑦=𝑥

Q7:

A dilation with center the origin is composed with a rotation about the origin to form a new linear transformation. The transformation formed sends the vector 34 to −3356.

Find the matrix representation of the transformation formed.

  • A512125
  • B5−12125
  • C−110014
  • D5−12125
  • E−12.921.441.4412.92

Find the scale factor of the original dilation.

  • Ascale factor = 169
  • Bscale factor = 13
  • Cscale factor = 154
  • Dscale factor = 13
  • Escale factor =√119

Q8:

The unit square, with vertices 𝑂(0,0), 𝐴(1,0), 𝐵(1,1), and 𝐶(0,1), is transformed by a rotation and then a dilation. Its image under this combined transformation is 𝑂𝐴𝐵𝐶, as shown in the diagram.

What are the coordinates of 𝐴′?

  • A23,23
  • B32,32
  • C32,−32
  • D23,−23
  • E32,32

What is the matrix of the combined transformation?

  • A⎡⎢⎢⎣32−32−3233⎤⎥⎥⎦
  • B⎡⎢⎢⎣3232−3232⎤⎥⎥⎦
  • C⎡⎢⎢⎣23−232323⎤⎥⎥⎦
  • D⎡⎢⎢⎣32−323232⎤⎥⎥⎦
  • E⎡⎢⎢⎣23−23−2323⎤⎥⎥⎦

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